On some curvature conditions of pseudosymmetry type
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On some curvature conditions of pseudosymmetry type Ryszard Deszcz · Małgorzata Głogowska · Marian Hotlo´s · Georges Zafindratafa
© The Author(s) 2015. This article is published with open access at Springerlink.com
Abstract It is known that the difference tensor R·C −C · R and the Tachibana tensor Q(S, C) of any semi-Riemannian Einstein manifold (M, g) of dimension n ≥ 4 are linearly dependent at every point of M. More precisely R · C − C · R = (1/(n − 1)) Q(S, C) holds on M. In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies R · C = C · R = Q(S, C) = 0. Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors R · C − C · R and Q(S, C) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property. Keywords Einstein manifold · Quasi-Einstein manifold · Manifold with parallel Weyl tensor · Roter type manifold · Pseudosymmetric manifold · Generalized Einstein metric condition · Tachibana tensor
R. Deszcz (B) · M. Głogowska Department of Mathematics, Wrocław University of Environmental and Life Sciences, Grunwaldzka 53, 50-357 Wrocław, Poland e-mail: [email protected] M. Głogowska e-mail: [email protected] M. Hotlo´s Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wrocław, Poland e-mail: [email protected] G. Zafindratafa Laboratoire de Mathématiques et Applications de Valenciennes, Université de Valenciennes et du Hainaut-Cambrésis, Le Mont Houy, Lamath-ISTV2, 59313 Valenciennes Cedex 9, France e-mail: [email protected]
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Mathematics Subject Classification Secondary 53C25
Primary 53B20 · 53B25 · 53B30 · 53B50;
1 Introduction Let ∇, R, S, κ and C be the Levi-Civita connection, the Riemann–Christoffel curvature tensor, the Ricci tensor, the scalar curvature tensor and the Weyl conformal curvature tensor of a semi-Riemannian manifold (M, g), n = dim M ≥ 2, respectively. It is well-known that the manifold (M, g), n ≥ 3, is said to be an Einstein manifold ([1]) if at every point of M its Ricci tensor S is proportional to the metric tensor g, i.e., S = κn g on M. In particular, if S vanishes on M then it is called Ricci flat. We denote by U S the set of all points of (M, g) at which S is not proportional to g, i.e., U S := {x ∈ M | S − κn g = 0 at x}. The manifold (M, g), n ≥ 3, is said to be a quasi-Einstein manifold if at every point x ∈ U S we have rank (S − α g) = 1, for some α ∈ R, i.e., S = α g + ε w ⊗ w, for some α ∈ R, where w is a non-zero covector at x and ε = ±1. We menti
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